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Theorem frgrncvvdeqlem1 30096
Description: Lemma 1 for frgrncvvdeq 30106. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem1 (𝜑𝑋𝑁)

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
2 df-nel 3042 . . . . 5 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
3 frgrncvvdeq.nx . . . . . 6 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2820 . . . . 5 (𝑌𝐷𝑌 ∈ (𝐺 NeighbVtx 𝑋))
52, 4xchbinx 334 . . . 4 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
61, 5sylib 217 . . 3 (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7 nbgrsym 29163 . . 3 (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
86, 7sylnibr 329 . 2 (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9 frgrncvvdeq.ny . . . 4 𝑁 = (𝐺 NeighbVtx 𝑌)
10 neleq2 3048 . . . 4 (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
119, 10ax-mp 5 . . 3 (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12 df-nel 3042 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
1311, 12bitri 275 . 2 (𝑋𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
148, 13sylibr 233 1 (𝜑𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1534  wcel 2099  wne 2935  wnel 3041  {cpr 4626  cmpt 5225  cfv 6542  crio 7369  (class class class)co 7414  Vtxcvtx 28796  Edgcedg 28847   NeighbVtx cnbgr 29132   FriendGraph cfrgr 30055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-nbgr 29133
This theorem is referenced by:  frgrncvvdeqlem7  30102  frgrncvvdeqlem8  30103  frgrncvvdeqlem9  30104
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